# Lesson 2.1: Quadratic Functions

Size: px
Start display at page:

## Transcription

1 Quadratic Functions: Lesson 2.1: Quadratic Functions Standard form (vertex form) of a quadratic function: Vertex: (h, k) Algebraically: *Use completing the square to convert a quadratic equation into standard quadratic function form. *Use evaluation and solve for the missing value of a when given a vertex and a point. How does the vertex of a parabola relate to the Maximum or Minimum of a quadratic function? Is it possible to find the vertex of a parabola written in general form without completing the square? For each function find the (a)vertex (b) x & y intercepts (c) zeros (d) sketch it.

2 Write the following parabola in vertex form: Vertex (3, -2) passes through (5,7) Vertex (-1,2); passes through (4,4) #62, #64 True or False: Without using your calculator, do the graphs of and have the same axis of symmetry.

3 Lesson 2.2: Polynomial Functions of Higher Degree *Graphs of polynomial functions are continuous, meaning it has no breaks. *Graphs of polynomial functions only have smooth rounded turns as opposed to sharp pointed turns. Polynomial function: When n is odd: END BEHAVIOR: A graph with a positive leading coefficient falls to the left and rises to the right. Left end-behavior: Right end-behavior: A graph with a negative leading coefficient rises to the left and falls to the right. Left end-behavior: Right end-behavior: When n is even: A graph with a positive leading coefficient rises to the left and right. Left end-behavior: Right end-behavior: A graph with a negative leading coefficient falls to the left and right. Left end-behavior: Right end-behavior: The Intermediate Value theorem: Let a and b be real numbers such that a < b. If f is a polynomial function such that f(a) f(b), then in the interval [a,b], f takes on every value between f(a) and f(b).

4 Polynomial function: The graph of f has at most n real zeros. The function f has at most (n 1) relative extrema. If f is a polynomial function and a is a real number: 1. x = a is a zero of the function f. 2. x = a is a solution of the polynomial equation f(x) = (x a) is a factor of the polynomial f(x). 4. (a, 0) is an x-intercept of the graph of f. In general, a factor of yields a repeated zero x = a of multiplicity k: If k is odd, the graph crosses the x-axis at x = a If k is even, the graph touches (but does not cross) the x-axis at x = a Find all the real zeros of the polynomial functions Find a polynomial function that has the given zeros. 2, -6 0,2,5 Can you solve these equations? From which polynomial (simplest) did we come from?

5 Analyze the given functions providing the x&y intercepts, zeros, action at the zeros, endbehavior. Then use your calculator to find the relative maximums & minimums, and give the increasing & decreasing intervals. #86, #88

6 Lesson 2.3: Real Zeros of Polynomial Functions Division Algorithm: Dividend = Divisor*Quotient + Remainder Long Division: Synthetic Division (when the divisor is in the form of x k) The Remainder Theorem: If a polynomial f(x) is divided by x k, the remainder is r = f(k) Use the remainder theorem to find the remainder. The Factor Theorem: A polynomial f(x) has a factor (x k) if and only if f(k) = 0 Is (x-1) a factor of the following function? f (x) = 2x 3 + 3x 2 8x + 3 The Rational Zero Test: see Pg. 118 List the possible rational zeros of f, use a calculator to narrow the list down, find the zeros. Given one real zero of a function find the others. ; x = ½

7 Lesson 2.4: Complex Numbers Complex number in standard form: a + bi, for real numbers a and b Conjugate: 4+6i -6 10i Complex plane: Simplify:

8 Lesson 2.5: The Fundamental Theorem of Algebra Linear Factorization Theorem: If f(x) is a polynomial of degree n, where n > 0, f has precisely n linear factors where are complex numbers and is the leading coefficient of f(x) If a + bi, where b 0, is a zero of a function, then the conjugate a bi is also a zero. Find all the zeros of the function and write the polynomial as a product of linear factors. Find the polynomial function with integer coefficients that has the given zeros. 2, 4 + i, 4 i 2,2,2,4i, -4i Use the given zero to find all the zeros of the function. ; zero: r = 2i ; zero:

9 Lesson 2.6: Rational Functions and Asymptotes Rational Function:, p(x) & q(x) are polynomials and q(x) isn t the zero polynomial. Horizontal Asymptote The line y = b is a horizontal asymptote of the graph of a function y = f(x) if either Vertical Asymptote The line x = a is a vertical asymptote of the graph of a function y = f(x) if either ; where p(x) and q(x) have no common factors. 1. The graph of f has vertical asymptotes at the zeros of q(x) 2. The graph of f has at most one horizontal asymptote, as follows a. If n < m, the x-axis (y = 0) is a horizontal asymptote b. If n = m, the line is a horizontal asymptote c. If n > m, the graph of f has no horizontal asymptote Analyzing graphs of rational functions 1.The y-int (if any) is the value of f(0) 2. The x-ints (if any) are the zeros of the numerator 3. The vert. asy.(if any) are the zeros of the denominator 4. The horizontal asy.: see a,b,and c above * It s all about the degree! Find the domain, and identify any horizontal and vertical asymptotes

10 f (x) = x 5 x Find the zeros (if any) of the rational function. Use a graphing utility to verify your answer. g(x) = x 2 4 x + 3 h(x) = x 2 +1 Write a rational function f having the specified characteristics. (There are many correct answers.) Vertical asymptotes: x = -2, x = 1 Vertical asymptotes: x = 0, x = 2.5 x = 5 Horizontal asymptotes: y = -3

11 Lesson 2.6: Rational Functions and Asymptotes Rational Function:, p(x) & q(x) are polynomials and q(x) isn t the zero polynomial. Horizontal Asymptote The line y = b is a horizontal asymptote of the graph of a function y = f(x) if either Vertical Asymptote The line x = a is a vertical asymptote of the graph of a function y = f(x) if either ; where p(x) and q(x) have no common factors. 1. The graph of f has vertical asymptotes at the zeros of q(x) 2. The graph of f has at most one horizontal asymptote, as follows a. If n < m, the x-axis (y = 0) is a horizontal asymptote b. If n = m, the line is a horizontal asymptote c. If n > m, the graph of f has no horizontal asymptote Analyzing graphs of rational functions 1.The y-int (if any) is the value of f(0) 2. The x-ints (if any) are the zeros of the numerator 3. The vert. asy.(if any) are the zeros of the denominator 4. The horizontal asy.: see a,b,and c above * It s all about the degree! Find the domain, and identify any horizontal and vertical asymptotes

12 f (x) = x 5 x Find the zeros (if any) of the rational function. Use a graphing utility to verify your answer. g(x) = x 2 4 x + 3 h(x) = x 2 +1 Write a rational function f having the specified characteristics. (There are many correct answers.) Vertical asymptotes: x = -2, x = 1 Vertical asymptotes: x = 0, x = 2.5 x = 5 Horizontal asymptotes: y = -3

13 Lesson 2.7: Graphs of rational functions If the degree of the numerator is exactly 1 more than the degree of the denominator, the graph of the rational function has a slant asymptote *use division to identify the slant asymptote. Find the intercepts and asymptotes to help you sketch the given rational functions.

### Chapter 2 Formulas and Definitions:

Chapter 2 Formulas and Definitions: (from 2.1) Definition of Polynomial Function: Let n be a nonnegative integer and let a n,a n 1,...,a 2,a 1,a 0 be real numbers with a n 0. The function given by f (x)

### Chapter 2 Polynomial and Rational Functions

Chapter 2 Polynomial and Rational Functions Section 1 Section 2 Section 3 Section 4 Section 5 Section 6 Section 7 Quadratic Functions Polynomial Functions of Higher Degree Real Zeros of Polynomial Functions

### Polynomial Functions and Models

1 CA-Fall 2011-Jordan College Algebra, 4 th edition, Beecher/Penna/Bittinger, Pearson/Addison Wesley, 2012 Chapter 4: Polynomial Functions and Rational Functions Section 4.1 Polynomial Functions and Models

### MAT116 Final Review Session Chapter 3: Polynomial and Rational Functions

MAT116 Final Review Session Chapter 3: Polynomial and Rational Functions Quadratic Function A quadratic function is defined by a quadratic or second-degree polynomial. Standard Form f x = ax 2 + bx + c,

### Polynomial and Rational Functions. Chapter 3

Polynomial and Rational Functions Chapter 3 Quadratic Functions and Models Section 3.1 Quadratic Functions Quadratic function: Function of the form f(x) = ax 2 + bx + c (a, b and c real numbers, a 0) -30

### Chapter 2: Polynomial and Rational Functions

Chapter 2: Polynomial and Rational Functions Section 2.1 Quadratic Functions Date: Example 1: Sketching the Graph of a Quadratic Function a) Graph f(x) = 3 1 x 2 and g(x) = x 2 on the same coordinate plane.

### Mission 1 Simplify and Multiply Rational Expressions

Algebra Honors Unit 6 Rational Functions Name Quest Review Questions Mission 1 Simplify and Multiply Rational Expressions 1) Compare the two functions represented below. Determine which of the following

### The Graph of a Quadratic Function. Quadratic Functions & Models. The Graph of a Quadratic Function. The Graph of a Quadratic Function

8/1/015 The Graph of a Quadratic Function Quadratic Functions & Models Precalculus.1 The Graph of a Quadratic Function The Graph of a Quadratic Function All parabolas are symmetric with respect to a line

### Lesson 7.1 Polynomial Degree and Finite Differences

Lesson 7.1 Polynomial Degree and Finite Differences 1. Identify the degree of each polynomial. a. 3x 4 2x 3 3x 2 x 7 b. x 1 c. 0.2x 1.x 2 3.2x 3 d. 20 16x 2 20x e. x x 2 x 3 x 4 x f. x 2 6x 2x 6 3x 4 8

### Table of contents. Polynomials Quadratic Functions Polynomials Graphs of Polynomials Polynomial Division Finding Roots of Polynomials

Table of contents Quadratic Functions Graphs of Polynomial Division Finding Roots of Jakayla Robbins & Beth Kelly (UK) Precalculus Notes Fall 2010 1 / 65 Concepts Quadratic Functions The Definition of

### n The coefficients a i are real numbers, n is a whole number. The domain of any polynomial is R.

Section 4.1: Quadratic Functions Definition: A polynomial function has the form P ( x ) = a x n+ a x n 1+... + a x 2+ a x + a (page 326) n n 1 2 1 0 The coefficients a i are real numbers, n is a whole

### Section Properties of Rational Expressions

88 Section. - Properties of Rational Expressions Recall that a rational number is any number that can be written as the ratio of two integers where the integer in the denominator cannot be. Rational Numbers:

### . As x gets really large, the last terms drops off and f(x) ½x

Pre-AP Algebra 2 Unit 8 -Lesson 3 End behavior of rational functions Objectives: Students will be able to: Determine end behavior by dividing and seeing what terms drop out as x Know that there will be

### Topic 25: Quadratic Functions (Part 1) A quadratic function is a function which can be written as 2. Properties of Quadratic Functions

Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 1 of 3 Topic 5: Quadratic Functions (Part 1) Definition: A quadratic function is a function which can be written as f x ax bx

### Polynomial Functions. Linear Graphs and Linear Functions 1.3

Polynomial Functions Linear Graphs and Linear Functions 1.3 Forms for equations of lines (linear functions) Ax + By = C Standard Form y = mx +b Slope-Intercept (y y 1 ) = m(x x 1 ) Point-Slope x = a Vertical

### Final Exam C Name i D) 2. Solve the equation by factoring. 4) x2 = x + 72 A) {1, 72} B) {-8, 9} C) {-8, -9} D) {8, 9} 9 ± i

Final Exam C Name First, write the value(s) that make the denominator(s) zero. Then solve the equation. 7 ) x + + 3 x - = 6 (x + )(x - ) ) A) No restrictions; {} B) x -, ; C) x -; {} D) x -, ; {2} Add

### H-Pre-Calculus Targets Chapter I can write quadratic functions in standard form and use the results to sketch graphs of the function.

H-Pre-Calculus Targets Chapter Section. Sketch and analyze graphs of quadratic functions.. I can write quadratic functions in standard form and use the results to sketch graphs of the function. Identify

### Power and Polynomial Functions. College Algebra

Power and Polynomial Functions College Algebra Power Function A power function is a function that can be represented in the form f x = kx % where k and p are real numbers, and k is known as the coefficient.

### 1) The line has a slope of ) The line passes through (2, 11) and. 6) r(x) = x + 4. From memory match each equation with its graph.

Review Test 2 Math 1314 Name Write an equation of the line satisfying the given conditions. Write the answer in standard form. 1) The line has a slope of - 2 7 and contains the point (3, 1). Use the point-slope

### 2 the maximum/minimum value is ( ).

Math 60 Ch3 practice Test The graph of f(x) = 3(x 5) + 3 is with its vertex at ( maximum/minimum value is ( ). ) and the The graph of a quadratic function f(x) = x + x 1 is with its vertex at ( the maximum/minimum

### Final Exam A Name. 20 i C) Solve the equation by factoring. 4) x2 = x + 30 A) {-5, 6} B) {5, 6} C) {1, 30} D) {-5, -6} -9 ± i 3 14

Final Exam A Name First, write the value(s) that make the denominator(s) zero. Then solve the equation. 1 1) x + 3 + 5 x - 3 = 30 (x + 3)(x - 3) 1) A) x -3, 3; B) x -3, 3; {4} C) No restrictions; {3} D)

### Chapter 7 Polynomial Functions. Factoring Review. We will talk about 3 Types: ALWAYS FACTOR OUT FIRST! Ex 2: Factor x x + 64

Chapter 7 Polynomial Functions Factoring Review We will talk about 3 Types: 1. 2. 3. ALWAYS FACTOR OUT FIRST! Ex 1: Factor x 2 + 5x + 6 Ex 2: Factor x 2 + 16x + 64 Ex 3: Factor 4x 2 + 6x 18 Ex 4: Factor

### Chapter 2 Polynomial and Rational Functions

Chapter 2 Polynomial and Rational Functions Overview: 2.2 Polynomial Functions of Higher Degree 2.3 Real Zeros of Polynomial Functions 2.4 Complex Numbers 2.5 The Fundamental Theorem of Algebra 2.6 Rational

### PreCalculus Notes. MAT 129 Chapter 5: Polynomial and Rational Functions. David J. Gisch. Department of Mathematics Des Moines Area Community College

PreCalculus Notes MAT 129 Chapter 5: Polynomial and Rational Functions David J. Gisch Department of Mathematics Des Moines Area Community College September 2, 2011 1 Chapter 5 Section 5.1: Polynomial Functions

### Polynomial functions right- and left-hand behavior (end behavior):

Lesson 2.2 Polynomial Functions For each function: a.) Graph the function on your calculator Find an appropriate window. Draw a sketch of the graph on your paper and indicate your window. b.) Identify

### Section 4.1: Polynomial Functions and Models

Section 4.1: Polynomial Functions and Models Learning Objectives: 1. Identify Polynomial Functions and Their Degree 2. Graph Polynomial Functions Using Transformations 3. Identify the Real Zeros of a Polynomial

### Math 1310 Section 4.1: Polynomial Functions and Their Graphs. A polynomial function is a function of the form ...

Math 1310 Section 4.1: Polynomial Functions and Their Graphs A polynomial function is a function of the form... where 0,,,, are real numbers and n is a whole number. The degree of the polynomial function

### Math Analysis Chapter 2 Notes: Polynomial and Rational Functions

Math Analysis Chapter Notes: Polynomial and Rational Functions Day 13: Section -1 Comple Numbers; Sections - Quadratic Functions -1: Comple Numbers After completing section -1 you should be able to do

### More Polynomial Equations Section 6.4

MATH 11009: More Polynomial Equations Section 6.4 Dividend: The number or expression you are dividing into. Divisor: The number or expression you are dividing by. Synthetic division: Synthetic division

### Chapter Five Notes N P U2C5

Chapter Five Notes N P UC5 Name Period Section 5.: Linear and Quadratic Functions with Modeling In every math class you have had since algebra you have worked with equations. Most of those equations have

### Operations w/polynomials 4.0 Class:

Exponential LAWS Review NO CALCULATORS Name: Operations w/polynomials 4.0 Class: Topic: Operations with Polynomials Date: Main Ideas: Assignment: Given: f(x) = x 2 6x 9 a) Find the y-intercept, the equation

### College Algebra Notes

Metropolitan Community College Contents Introduction 2 Unit 1 3 Rational Expressions........................................... 3 Quadratic Equations........................................... 9 Polynomial,

### Chapter 3: Polynomial and Rational Functions

Chapter 3: Polynomial and Rational Functions 3.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P (x) = a n x n + a n 1 x n 1 + + a 1 x + a 0 The numbers

2 Polynomial and Rational Functions Copyright Cengage Learning. All rights reserved. 2.2 Polynomial Functions of Higher Degree Copyright Cengage Learning. All rights reserved. What You Should Learn Use

### Chapter 3 Polynomial Functions

Trig / Coll. Alg. Name: Chapter 3 Polynomial Functions 3.1 Quadratic Functions (not on this test) For each parabola, give the vertex, intercepts (x- and y-), axis of symmetry, and sketch the graph. 1.

### 6x 3 12x 2 7x 2 +16x 7x 2 +14x 2x 4

2.3 Real Zeros of Polynomial Functions Name: Pre-calculus. Date: Block: 1. Long Division of Polynomials. We have factored polynomials of degree 2 and some specific types of polynomials of degree 3 using

### 3 Polynomial and Rational Functions

3 Polynomial and Rational Functions 3.1 Polynomial Functions and their Graphs So far, we have learned how to graph polynomials of degree 0, 1, and. Degree 0 polynomial functions are things like f(x) =,

### (a) Write down the value of q and of r. (2) Write down the equation of the axis of symmetry. (1) (c) Find the value of p. (3) (Total 6 marks)

1. Let f(x) = p(x q)(x r). Part of the graph of f is shown below. The graph passes through the points ( 2, 0), (0, 4) and (4, 0). (a) Write down the value of q and of r. (b) Write down the equation of

### Cumulative Review. Name. 13) 2x = -4 13) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Cumulative Review Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Evaluate the algebraic expression for the given value or values of the variable(s).

### 2.1 Quadratic Functions

Date:.1 Quadratic Functions Precalculus Notes: Unit Polynomial Functions Objective: The student will sketch the graph of a quadratic equation. The student will write the equation of a quadratic function.

### Making Connections with Rational Functions and Equations

Section 3.5 Making Connections with Rational Functions and Equations When solving a problem, it's important to read carefully to determine whether a function is being analyzed (Finding key features) or

### 171S4.3 Polynomial Division; The Remainder and Factor Theorems. October 26, Polynomial Division; The Remainder and Factor Theorems

MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial

### 171S4.3 Polynomial Division; The Remainder and Factor Theorems. March 24, Polynomial Division; The Remainder and Factor Theorems

MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial

### Math 120, Sample Final Fall 2015

Math 10, Sample Final Fall 015 Disclaimer: This sample final is intended to help students prepare for the final exam The final exam will be similar in structure and type of problems, however the actual

### How many solutions are real? How many solutions are imaginary? What are the solutions? (List below):

1 Algebra II Chapter 5 Test Review Standards/Goals: F.IF.7.c: I can identify the degree of a polynomial function. F.1.a./A.APR.1.: I can evaluate and simplify polynomial expressions and equations. F.1.b./

### Section 3.1 Quadratic Functions

Chapter 3 Lecture Notes Page 1 of 72 Section 3.1 Quadratic Functions Objectives: Compare two different forms of writing a quadratic function Find the equation of a quadratic function (given points) Application

### Chapter 8. Exploring Polynomial Functions. Jennifer Huss

Chapter 8 Exploring Polynomial Functions Jennifer Huss 8-1 Polynomial Functions The degree of a polynomial is determined by the greatest exponent when there is only one variable (x) in the polynomial Polynomial

### Section 2: Polynomial and Rational Functions

Section 2: Polynomial and Rational Functions The following maps the videos in this section to the Texas Essential Knowledge and Skills for Mathematics TAC 111.42(c). 2.01 Quadratic Functions Precalculus

### Lesson 19 Factoring Polynomials

Fast Five Lesson 19 Factoring Polynomials Factor the number 38,754 (NO CALCULATOR) Divide 72,765 by 38 (NO CALCULATOR) Math 2 Honors - Santowski How would you know if 145 was a factor of 14,436,705? What

### MATH 121: EXTRA PRACTICE FOR TEST 2. Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam.

MATH 121: EXTRA PRACTICE FOR TEST 2 Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam. 1 Linear Functions 1. Consider the functions f(x) = 3x + 5 and g(x)

### Lesson 7.1 Polynomial Degree and Finite Differences

Lesson 7.1 Polynomial Degree and Finite Differences 1. Identify the degree of each polynomial. a. 1 b. 0.2 1. 2 3.2 3 c. 20 16 2 20 2. Determine which of the epressions are polynomials. For each polynomial,

### Section 0.2 & 0.3 Worksheet. Types of Functions

MATH 1142 NAME Section 0.2 & 0.3 Worksheet Types of Functions Now that we have discussed what functions are and some of their characteristics, we will explore different types of functions. Section 0.2

### ZEROS OF POLYNOMIAL FUNCTIONS ALL I HAVE TO KNOW ABOUT POLYNOMIAL FUNCTIONS

ZEROS OF POLYNOMIAL FUNCTIONS ALL I HAVE TO KNOW ABOUT POLYNOMIAL FUNCTIONS TOOLS IN FINDING ZEROS OF POLYNOMIAL FUNCTIONS Synthetic Division and Remainder Theorem (Compressed Synthetic Division) Fundamental

### All quadratic functions have graphs that are U -shaped and are called parabolas. Let s look at some parabolas

Chapter Three: Polynomial and Rational Functions 3.1: Quadratic Functions Definition: Let a, b, and c be real numbers with a 0. The function f (x) = ax 2 + bx + c is called a quadratic function. All quadratic

### Unit 4 Polynomial/Rational Functions Zeros of Polynomial Functions (Unit 4.3)

Unit 4 Polynomial/Rational Functions Zeros of Polynomial Functions (Unit 4.3) William (Bill) Finch Mathematics Department Denton High School Lesson Goals When you have completed this lesson you will: Find

### Unit 1: Polynomial Functions SuggestedTime:14 hours

Unit 1: Polynomial Functions SuggestedTime:14 hours (Chapter 3 of the text) Prerequisite Skills Do the following: #1,3,4,5, 6a)c)d)f), 7a)b)c),8a)b), 9 Polynomial Functions A polynomial function is an

### Review all the activities leading to Midterm 3. Review all the problems in the previous online homework sets (8+9+10).

MA109, Activity 34: Review (Sections 3.6+3.7+4.1+4.2+4.3) Date: Objective: Additional Assignments: To prepare for Midterm 3, make sure that you can solve the types of problems listed in Activities 33 and

Student: Date: Instructor: kumnit nong Course: MATH 105 by Nong https://xlitemprodpearsoncmgcom/api/v1/print/math Assignment: CH test review 1 Find the transformation form of the quadratic function graphed

### SB CH 2 answers.notebook. November 05, Warm Up. Oct 8 10:36 AM. Oct 5 2:22 PM. Oct 8 9:22 AM. Oct 8 9:19 AM. Oct 8 9:26 AM.

Warm Up Oct 8 10:36 AM Oct 5 2:22 PM Linear Function Qualities Oct 8 9:22 AM Oct 8 9:19 AM Quadratic Function Qualities Oct 8 9:26 AM Oct 8 9:25 AM 1 Oct 8 9:28 AM Oct 8 9:25 AM Given vertex (-1,4) and

Warm-ups 1 2.2 Polynomial Functions of Higher Degree Copyright Cengage Learning. All rights reserved. Objectives Use transformations to sketch graphs of polynomial functions. Use the Leading Coefficient

### To get horizontal and slant asymptotes algebraically we need to know about end behaviour for rational functions.

Concepts: Horizontal Asymptotes, Vertical Asymptotes, Slant (Oblique) Asymptotes, Transforming Reciprocal Function, Sketching Rational Functions, Solving Inequalities using Sign Charts. Rational Function

### Polynomial and Rational Functions

Name Date Chapter Polnomial and Rational Functions Section.1 Quadratic Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Important Vocabular Define

### 3.1 Power Functions & Polynomial Functions

3.1 Power Functions & Polynomial Functions A power function is a function that can be represented in the form f() = p, where the base is a variable and the eponent, p, is a number. The Effect of the Power

### Polynomials. Exponents. End Behavior. Writing. Solving Factoring. Graphing. End Behavior. Polynomial Notes. Synthetic Division.

Polynomials Polynomials 1. P 1: Exponents 2. P 2: Factoring Polynomials 3. P 3: End Behavior 4. P 4: Fundamental Theorem of Algebra Writing real root x= 10 or (x+10) local maximum Exponents real root x=10

### Modeling Data. 27 will get new packet. 24 Mixed Practice 3 Binomial Theorem. 23 Fundamental Theorem March 2

Name: Period: Pre-Cal AB: Unit 1: Polynomials Monday Tuesday Block Friday 11/1 1 Unit 1 TEST Function Operations and Finding Inverses 16 17 18/19 0 NO SCHOOL Polynomial Division Roots, Factors, Zeros and

### Mock Final Exam Name. Solve and check the linear equation. 1) (-8x + 8) + 1 = -7(x + 3) A) {- 30} B) {- 6} C) {30} D) {- 28}

Mock Final Exam Name Solve and check the linear equation. 1) (-8x + 8) + 1 = -7(x + 3) 1) A) {- 30} B) {- 6} C) {30} D) {- 28} First, write the value(s) that make the denominator(s) zero. Then solve the

### A repeated root is a root that occurs more than once in a polynomial function.

Unit 2A, Lesson 3.3 Finding Zeros Synthetic division, along with your knowledge of end behavior and turning points, can be used to identify the x-intercepts of a polynomial function. This information allows

### Dividing Polynomials: Remainder and Factor Theorems

Dividing Polynomials: Remainder and Factor Theorems When we divide one polynomial by another, we obtain a quotient and a remainder. If the remainder is zero, then the divisor is a factor of the dividend.

### Calculus First Semester Review Name: Section: Evaluate the function: (g o f )( 2) f (x + h) f (x) h. m(x + h) m(x)

Evaluate the function: c. (g o f )(x + 2) d. ( f ( f (x)) 1. f x = 4x! 2 a. f( 2) b. f(x 1) c. f (x + h) f (x) h 4. g x = 3x! + 1 Find g!! (x) 5. p x = 4x! + 2 Find p!! (x) 2. m x = 3x! + 2x 1 m(x + h)

### Advanced Algebra II 1 st Semester Exam Review

dname Advanced Algebra II 1 st Semester Exam Review Chapter 1A: Number Sets & Solving Equations Name the sets of numbers to which each number belongs. 1. 34 2. 525 3. 0.875 4. Solve each equation. Check

### Skills Practice Skills Practice for Lesson 10.1

Skills Practice Skills Practice for Lesson.1 Name Date Higher Order Polynomials and Factoring Roots of Polynomial Equations Problem Set Solve each polynomial equation using factoring. Then check your solution(s).

### Math 1314 Lesson 1: Prerequisites. Example 1: Simplify and write the answer without using negative exponents:

Math 1314 Lesson 1: Prerequisites 1. Exponents 1 m n n n m Recall: x = x = x n x Example 1: Simplify and write the answer without using negative exponents: a. x 5 b. ( x) 5 Example : Write as a radical:

### MTH30 Review Sheet. y = g(x) BRONX COMMUNITY COLLEGE of the City University of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE

BRONX COMMUNITY COLLEGE of the City University of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE MTH0 Review Sheet. Given the functions f and g described by the graphs below: y = f(x) y = g(x) (a)

### King Fahd University of Petroleum and Minerals Prep-Year Math Program Math Term 161 Recitation (R1, R2)

Math 001 - Term 161 Recitation (R1, R) Question 1: How many rational and irrational numbers are possible between 0 and 1? (a) 1 (b) Finite (c) 0 (d) Infinite (e) Question : A will contain how many elements

### Introduction. A rational function is a quotient of polynomial functions. It can be written in the form

RATIONAL FUNCTIONS Introduction A rational function is a quotient of polynomial functions. It can be written in the form where N(x) and D(x) are polynomials and D(x) is not the zero polynomial. 2 In general,

### Let's look at some higher order equations (cubic and quartic) that can also be solved by factoring.

GSE Advanced Algebra Polynomial Functions Polynomial Functions Zeros of Polynomial Function Let's look at some higher order equations (cubic and quartic) that can also be solved by factoring. In the video,

### Chapter 2. Polynomial and Rational Functions. 2.6 Rational Functions and Their Graphs. Copyright 2014, 2010, 2007 Pearson Education, Inc.

Chapter Polynomial and Rational Functions.6 Rational Functions and Their Graphs Copyright 014, 010, 007 Pearson Education, Inc. 1 Objectives: Find the domains of rational functions. Use arrow notation.

### The final is cumulative, but with more emphasis on chapters 3 and 4. There will be two parts.

Math 141 Review for Final The final is cumulative, but with more emphasis on chapters 3 and 4. There will be two parts. Part 1 (no calculator) graphing (polynomial, rational, linear, exponential, and logarithmic

### Pre-Algebra 2. Unit 9. Polynomials Name Period

Pre-Algebra Unit 9 Polynomials Name Period 9.1A Add, Subtract, and Multiplying Polynomials (non-complex) Explain Add the following polynomials: 1) ( ) ( ) ) ( ) ( ) Subtract the following polynomials:

### 2.6. Graphs of Rational Functions. Copyright 2011 Pearson, Inc.

2.6 Graphs of Rational Functions Copyright 2011 Pearson, Inc. Rational Functions What you ll learn about Transformations of the Reciprocal Function Limits and Asymptotes Analyzing Graphs of Rational Functions

### Reading Mathematical Expressions & Arithmetic Operations Expression Reads Note

Math 001 - Term 171 Reading Mathematical Expressions & Arithmetic Operations Expression Reads Note x A x belongs to A,x is in A Between an element and a set. A B A is a subset of B Between two sets. φ

### Polynomial Expressions and Functions

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 1 - of 36 Topic 32: Polynomial Expressions and Functions Recall the definitions of polynomials and terms. Definition: A polynomial

### Semester Review Packet

MATH 110: College Algebra Instructor: Reyes Semester Review Packet Remarks: This semester we have made a very detailed study of four classes of functions: Polynomial functions Linear Quadratic Higher degree

### Pre-Calculus: Functions and Their Properties (Solving equations algebraically and graphically, matching graphs, tables, and equations, and

Pre-Calculus: 1.1 1.2 Functions and Their Properties (Solving equations algebraically and graphically, matching graphs, tables, and equations, and finding the domain, range, VA, HA, etc.). Name: Date:

### A Partial List of Topics: Math Spring 2009

A Partial List of Topics: Math 112 - Spring 2009 This is a partial compilation of a majority of the topics covered this semester and may not include everything which might appear on the exam. The purpose

### Section 5.1 Determine if a function is a polynomial function. State the degree of a polynomial function.

Test Instructions Objectives Section 5.1 Section 5.1 Determine if a function is a polynomial function. State the degree of a polynomial function. Form a polynomial whose zeros and degree are given. Graph

### 1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.

Math120 - Precalculus. Final Review. Fall, 2011 Prepared by Dr. P. Babaali 1 Algebra 1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.

3.2 Polynomial Functions and Their Graphs Copyright Cengage Learning. All rights reserved. Objectives Graphing Basic Polynomial Functions End Behavior and the Leading Term Using Zeros to Graph Polynomials

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. B) 6x + 4

Math1420 Review Comprehesive Final Assessment Test Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Add or subtract as indicated. x + 5 1) x2

### Chapter 2. Polynomial and Rational Functions. 2.3 Polynomial Functions and Their Graphs. Copyright 2014, 2010, 2007 Pearson Education, Inc.

Chapter Polynomial and Rational Functions.3 Polynomial Functions and Their Graphs Copyright 014, 010, 007 Pearson Education, Inc. 1 Objectives: Identify polynomial functions. Recognize characteristics

### 32. Use a graphing utility to find the equation of the line of best fit. Write the equation of the line rounded to two decimal places, if necessary.

Pre-Calculus A Final Review Part 2 Calculator Name 31. The price p and the quantity x sold of a certain product obey the demand equation: p = x + 80 where r = xp. What is the revenue to the nearest dollar

### Pre-Calculus Midterm Practice Test (Units 1 through 3)

Name: Date: Period: Pre-Calculus Midterm Practice Test (Units 1 through 3) Learning Target 1A I can describe a set of numbers in a variety of ways. 1. Write the following inequalities in interval notation.

### MATH 121: EXTRA PRACTICE FOR TEST 2. Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam.

MATH 121: EXTRA PRACTICE FOR TEST 2 Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam. 1 Linear Functions 1. Consider the functions f(x) = 3x + 5 and g(x)

### 10/22/16. 1 Math HL - Santowski SKILLS REVIEW. Lesson 15 Graphs of Rational Functions. Lesson Objectives. (A) Rational Functions

Lesson 15 Graphs of Rational Functions SKILLS REVIEW! Use function composition to prove that the following two funtions are inverses of each other. 2x 3 f(x) = g(x) = 5 2 x 1 1 2 Lesson Objectives! The

### Algebra 2 Honors: Final Exam Review

Name: Class: Date: Algebra 2 Honors: Final Exam Review Directions: You may write on this review packet. Remember that this packet is similar to the questions that you will have on your final exam. Attempt

### The degree of the polynomial function is n. We call the term the leading term, and is called the leading coefficient. 0 =

Math 1310 A polynomial function is a function of the form = + + +...+ + where 0,,,, are real numbers and n is a whole number. The degree of the polynomial function is n. We call the term the leading term,

### PreCalculus: Semester 1 Final Exam Review

Name: Class: Date: ID: A PreCalculus: Semester 1 Final Exam Review Short Answer 1. Determine whether the relation represents a function. If it is a function, state the domain and range. 9. Find the domain

### 11 /2 12 /2 13 /6 14 /14 15 /8 16 /8 17 /25 18 /2 19 /4 20 /8

MAC 1147 Exam #1a Answer Key Name: Answer Key ID# Summer 2012 HONOR CODE: On my honor, I have neither given nor received any aid on this examination. Signature: Instructions: Do all scratch work on the